Über die Eulersche ϕ-funktion in quadratischen Zahlkörpern

Let (a) denote the Euler totient function in an arbitrary quadratic number fieldK and defineE _K (x) andH _K (x) as the error terms in the asymptotic formulae for and , respectively, summation being extended over all ideals a with 1N(a)x. In this paper the asymptotic behaviour of _n=1 ^N E _K (n) and _n=1 ^N H _K (n)H _K (itn) is studied. This generalizes results ofPillai andChowla [5] on the classical case.

---

---

Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet     

On the addition of residue classes modp

Given distinct residue classesa ₁ ,...,a _k modulo a primep, we consider the setS _j of all sums , whereli ₁<...<i _j k. We give a sufficient condition for the inequality |S _j |j(k–j)+1 to hold.     

Chu-Dualität und zwei Klassen maximal fastperiodischer Gruppen

This is a continuation of the paper Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen, [6]. In [6], the classes and were introduced. We give sufficient conditions to conclude thatG is in if one knows thatG/G ₀ is in . If a groupG is in and ifG satisfies the Chu-duality then all closed subgroups ofG satisfy the Chu-duality. The Chu-quasi-dual of the Heisenberg groupH with integral coefficients is computed. It is shown thatH does not satisfy the Chu-duality, thatH is in , and thatH is not in .     

On Mercer's theorem for (C,2)-means

A direct proof is given for the following theorem, contained as a special case in a more general result ofPolniakowski:Theorem. Denote _n the Cesaro-means of order 2 of the sequenceS ₁, S₂, ... , any real number satisfying 0<<=1. Then implies To Prof. Th. Schneider on the occasion of his 65^th birthday     

New constructions for De Bruijn tori

A De Bruijn torus is a periodicd-dimensionalk-ary array such that eachn ₁ × ... ×n _d k-ary array appears exactly once with the same period. We describe two new methods of constructing such arrays. The first is a type of product that constructs ak ₁ k ₂ -ary torus from ak ₁ -ary torus and ak ₂ -ary torus. The second uses a decomposition of ad-dimensional torus to produce ad+1 dimensional torus. Both constructions will produce two dimensionalk-ary tori for which the period is not a power ofk. In particular, for and for all natural numbers (n ₁ , n ₂ ), we construct 2-dimensionalk-ary De Bruijn tori with order n ₁ , n ₂ and period where .Dedicated to the memory of Tony BrewsterPartially supported by NSF grant DMS-9201467Partially supported by a grant from the Reidler Foundation     

Sur les plus grands facteurs premiers d'un entier

Let and, for each integern such that (n)k, denote byP _k (n) itsk ^th largest prime factor. Further, given a set of primesQ of positive density <1 satisfying a certain regularity condition, defineP(n, Q), as the largest prime divisor ofn belonging toQ, assuming thatP(n,Q)=+ if no such prime factor exists. We provide estimates of , fork2, and of . We also study the median value of the functionP(n,Q) and that of the functionP _k (n) for eachk1.     

Factorizations of One-Generated Composition Formations

A non-empty formation of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group G having a normal subgroup N such that belongs to . An intersection of all composition formations containing a given group G is denoted cformG. Conditions are described under which has the form , where .     

On theq-log-concavity of Gaussian binomial coefficients

We give a combinatorial proof that is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l withab andlk. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients , which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775).     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

Measurable Hermitian-positive functions

Let , respectively, denote the sets of continuous, measurable, and almost-everywhere vanishing functions f(x) (–a<x<a; f(0)>0). The theorem is proved that for every there correspond and , such that f=f_c + f_s. Some unsolved problems related to this theorem are formulated.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 79–89, January, 1978.     

Homogeneous algebraic varieties defined by Jordan pairs

Every Jordan pair defines an algebraic varietyX containing as a dense open subset.X is projective (affine) if and only if is separable (radical). The Picard group ofX is generated by the irreducible factors of the generic norm of . If is separable then the automorphism group ofX is the projective group of .     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Abschätzungen für\sum\limits_{n \leqslant N} {B_1 (n\alpha )}

Estimates for . Let and letB ₁(x)={x}–1/2. In this paper we shall give best possible estimates for . On the importance of this sum see the papers ofBehnke [2],Hardy andLittlewood [5],Hecke [6] andOstrowski [9].

---

---

Meinem verehrten Lehrer, Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f, we employ a convex approximation f _{k + 1} at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.     

Functions of translation type and functorial properties of Segal algebras II

In this paper we investigate functorial properties of the Segal algebra which consists of all functionsf in Wiener's algebra onG with Fourier transform in Wiener's algebra on the dual group . Especially may serve as a very large and natural domain for Poisson's formula. Moreover, there is introduced a Segal algebraE ₀(G) containing as a subspace, but still eachfE ₀(S) satisfies Poisson's formula.     

Equational Theories for Classes of Finite Semigroups

It is proved that there exists an infinite sequence of finitely based semigroup varieties such that, for all i, an equational theory for and for the class of all finite semigroups in is undecidable while an equational theory for and for the class of all finite semigroups in is decidable. An infinite sequence of finitely based semigroup varieties is constructed so that, for all i, an equational theory for and for the class of all finite semigroups in is decidable whicle an equational theory for and for the class of all finite semigroups in is not.     

An Ω-theorem for an error term related to the sum-of-divisors function

Let denote the sum-of-divisors function, and set . Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)= (x log logx). In this paper we obtain the more preciseE ₁ (x)=_–(x log logx). The method consists in averaging over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions, .